The CUR decomposition is a factorization of a low-rank matrix obtained by selecting certain column and row submatrices of it. We perform a thorough investigation of what happens to such decompositions in the presence of noise. Since CUR decompositions are non-uniquely formed, we investigate several variants and give perturbation estimates for each in terms of the magnitude of the noise matrix in a broad class of norms which includes all Schatten p-norms. The estimates given here are qualitative and illustrate how the choice of columns and rows affects the quality of the approximation, and additionally we obtain new state-of-the-art bounds for some variants of CUR approximations.2010 Mathematics Subject Classification. 15A23,65F30,68P99,68W20.

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We investigate systems of the form {A t g : g ∈ G, t ∈ [0, L]} where A ∈ B(H) is a normal operator in a separable Hilbert space H, G ⊂ H is a countable set, and L is a positive real number. Although the main goal of this work is to study the frame properties of {A t g : g ∈ G, t ∈ [0, L]}, as intermediate steps, we explore the completeness and Bessel properties of such systems from a theoretical perspective, which are of interest by themselves. Beside the theoretical appeal of investigating such systems, their connections to dynamical and mobile sampling make them fundamental for understanding and solving several major problems in engineering and science.2010 Mathematics Subject Classification. 46N99, 42C15, 94O20.

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